Oblique decision trees recursively divide the feature space by using splits based on linear combinations of attributes. Compared to their univariate counterparts, which only use a single attribute per split, they are often smaller and more accurate. A common approach to learn decision trees is by iteratively introducing splits on a training set in a top–down manner, yet determining a single optimal oblique split is in general computationally intractable. Therefore, one has to rely on heuristics to find near-optimal splits. In this paper, we adapt the cross-entropy optimization method to tackle this problem. The approach is motivated geometrically by the observation that equivalent oblique splits can be interpreted as connected regions on a unit hypersphere which are defined by the samples in the training data. In each iteration, the algorithm samples multiple candidate solutions from this hypersphere using the von Mises–Fisher distribution which is parameterized by a mean direction and a concentration parameter. These parameters are then updated based on the best performing samples such that when the algorithm terminates a high probability mass is assigned to a region of near-optimal solutions. Our experimental results show that the proposed method is well-suited for the induction of compact and accurate oblique decision trees in a small amount of time.

Univariate decision tree induction methods for multiclass classification problems such as CART, C4.5 and ID3 continue to be very popular in the context of machine learning due to their major benefit of being easy to interpret. However, as these trees only consider a single attribute per node, they often get quite large which lowers their explanatory value. Oblique decision tree building algorithms, which divide the feature space by multidimensional hyperplanes, often produce much smaller trees but the individual splits are hard to interpret. Moreover, the effort of finding optimal oblique splits is very high such that heuristics have to be applied to determine local optimal solutions. In this work, we introduce an effective branch and bound procedure to determine global optimal bivariate oblique splits for concave impurity measures. Decision trees based on these bivariate oblique splits remain fairly interpretable due to the restriction to two attributes per split. The resulting trees are significantly smaller and more accurate than their univariate counterparts due to their ability of adapting better to the underlying data and capturing interactions of attribute pairs. Moreover, our evaluation shows that our algorithm even outperforms algorithms based on heuristically obtained multivariate oblique splits despite the fact that we are focusing on two attributes only.

Human-machine teams are a key component of future production systems, but formation of the teams is challenging due to uncertain and evolving skills and preferences. We propose a decentralized multi-round auction approach, where job agents specify processing times and workers bid their preferred job execution modes representing possible human-machine teams. The factory assigns jobs to teams while maximizing the number of jobs assigned within the specified planning period. Afterwards, a post-auction mechanism improves the assignment in a fair way without worsening the goal achievement of any party involved. We evaluate our auction and post-auction mechanisms by analyzing the workers' social welfare and the factory's objective for different bidding strategies. It turns out that our mechanism yields efficient assignments and provides incentives for the workers to share true estimations about their capabilities.

Many production systems still rely on human workers, who—unlike machines—have individual preferences and private information to be taken into account when assigning staff to jobs. We propose a multi-round auction where workers bid their work rate on jobs trying to receive the jobs they prefer. The workers use a bidding strategy based on the myopic best response rule. The factory assigns the jobs to the workers while trying to maximize its production rate. To cope with the multiplicity of optimal assignments, the workers apply the Hurwicz criterion combining the best and worst possible outcomes. We evaluate our auction mechanism by examining the workers’ utilities and the factory’s production rate on generated problem instances and compare the results to a similar smart market mechanism for procurement and to the well-known Vickrey-Clarke-Groves auction. Our auction allows the factory to receive information about the maximum work rate of the workers. In return, the workers participate in the job assignment and can incorporate their preferences.

The way humans and artificially intelligent machines interact is undergoing a dramatic change. This change becomes particularly apparent in domains where humans and machines collaboratively work on joint tasks or objects in teams, such as in industrial assembly or disassembly processes. While there is intensive research work on human–machine collaboration in different research disciplines, systematic and interdisciplinary approaches towards engineering systems that consist of or comprise human–machine teams are still rare. In this paper, we review and analyze the state of the art, and derive and discuss core requirements and concepts by means of an illustrating scenario. In terms of methods, we focus on how reciprocal trust between humans and intelligent machines is defined, built, measured, and maintained from a systems engineering and planning perspective in literature. Based on our analysis, we propose and outline three important areas of future research on engineering and operating human–machine teams for trusted collaboration. For each area, we describe exemplary research opportunities.

State-of-the-art decision tree algorithms are top-down induction heuristics which greedily partition the attribute space by iteratively choosing the best split on an individual attribute. Despite their attractive performance in terms of runtime, simple examples, such as the XOR-Problem, point out that these heuristics often fail to find the best classification rules if there are strong interactions between two or more attributes from the given datasets. In this context, we present a branch and bound based decision tree algorithm to identify optimal bivariate axis-aligned splits according to a given impurity measure. In contrast to a univariate split that can be found in linear time, such an optimal cross-split has to consider every combination of values for every possible selection of pairs of attributes which leads to a combinatorial optimization problem that is quadratic in the number of values and attributes. To overcome this complexity, we use a branch and bound procedure, a well known technique from combinatorial optimization, to divide the solution space into several sets and to detect the optimal cross-splits in a short amount of time. These cross splits can either be used directly to construct quaternary decision trees or they can be used to select only the better one of the individual splits. In the latter case, the outcome is a binary decision tree with a certain sense of foresight for correlated attributes. We test both of these variants on various datasets of the UCI Machine Learning Repository and show that cross-splits can consistently produce smaller decision trees than state-of-the-art methods with comparable accuracy. In some cases, our algorithm produces considerably more accurate trees due to the ability of drawing more elaborate decisions than univariate induction algorithms.

Platooning – vehicles travelling close together behaving as a unit – aims to improve network throughput both on highways and in urban traffic. We study the problem of platoon formation in an urban environment using the scenario of logistic service providers equipped with fleets of autonomously driving pods to carry out same-day delivery tasks by creating cross-provider platoons. The novelty of our work is that we investigate the problem of cross-provider platoons, i.e., platoons with members from different self-interested logistic service providers. Our aim is to study platoon formation mechanisms and possible benefits of cross-provider platooning using simulation. We formulate optimal platoon formation as an integer linear optimisation problem (ILP), aiming to find the longest sub-routes to be shared between vehicles by platooning. The proposed method was implemented and tested on a mesoscopic model to simulate platoon formation and operation, on real network data with realistic background traffic models. Comparing our method to a simpler route matching algorithm reveals comparable system level performance; however, our method performs better with respect to local participant utility, i.e.appears more suited to take vehicle/provider preferences into account.

We consider the classical mechanism design problem of fully implementing social choice functions in dominant strategies in settings where monetary payments are allowed and the utility functions are quasi-linear. We consider both the general question of full implementation by indirect mechanisms and the special case of full implementation by incentive compatible direct revelation mechanisms. For the general case of full implementation by indirect mechanisms, we prove that one can restrict attention to incentive compatible augmented revelation mechanisms, in which the type space of each agent is a subset of the set of her possible bids and truthful reporting is a dominant strategy equilibrium. When the type spaces of the agents are finite, we give a complete characterization of the set of social choice functions that can be fully implemented in dominant strategies. For the case that one restricts to incentive compatible direct revelation mechanisms, we show that an adaption of the well-known negative cycle criterion for partial implementability also characterizes the social choice functions that are fully implementable.

We consider the problem of scheduling intervals on m identical machines where each interval can be seen as a job with fixed start and end time. The goal is to accept a maximum cardinality subset of the given intervals and assign these intervals to the machines subject to the constraint that no two intervals assigned to the same machine overlap. We analyze an online version of this problem where, initially, a set of n potential intervals and an upper bound k on the number of failing intervals is given. If an interval fails, it can be accepted neither by the online algorithm nor by the adversary. An online algorithm learns that an interval fails at the time when it is supposed to be started. If a non-failing interval is accepted, it cannot be aborted and must be processed non-preemptively until completion. For different settings of this problem, we present deterministic and randomized online algorithms and prove lower bounds on the competitive ratio.

We consider an online knapsack problem with incremental capacity. In each time period, a set of items, each with a specific weight and value, is revealed and, without knowledge of future items, it has to be decided which of these items to accept. Additionally, the knapsack capacity is not fully available from the start but increases by a constant amount in each time period. The goal is to maximize the overall value of the accepted items. This setting extends the basic online knapsack problem by introducing a dynamic instead of a static knapsack capacity and is applicable to classic problems such as resource allocation or one-way trading. In contrast to the basic online knapsack problem, for which no competitive algorithms exist, the setting of incremental capacity facilitates the development of competitive algorithms for a bounded time horizon. We provide a competitive analysis of deterministic and randomized online algorithms for the online knapsack problem with incremental capacity and present lower bounds on the competitive ratio achievable by online algorithms for the problem. Most of these lower bounds match the competitive ratios achieved by our online algorithms exactly or differ only by a constant factor.

Unmanned Aerial Vecicles (UAVs) in civil and military applications are becoming increasingly popular. Various platform types have already shown their great potential in missions that require rapid surveillance capabilities or logistic support. Large scale incidents require the deployment of several platforms with various capabilities. In this case, coordinated use will lead to more efficient use of the given resources. Problems to resolve resemble known optimization problems from the field of vehicle routing or scheduling. The problem considered in this work includes a given team of homogenous UAVs and a set of target locations with certain requests that need to be served. It is modeled as a variant of the Vehicle Routing Problem (VRP) that is known to be NP hard, i.e. until now no algorithm is known that can solve the problem in polynomial run-time. In this paper, the problem is formulated using a path flow formulation and a column generation algorithm has been implemented and tested to solve simulated real-time instances of the problem in suitable time*.

The traveling tournament problem is a well-known combinatorial optimization problem with direct applications to sport leagues scheduling, that sparked intensive algorithmic research over the last decade. With the Challenge Traveling Tournament Instances as an established benchmark, the most successful approaches to the problem use meta-heuristics like tabu search or simulated annealing, partially heavily parallelized. Integer programming based methods on the other hand are hardly able to tackle larger benchmark instances. In this work we present a hybrid approach that draws on the power of commercial integer programming solvers as well as the speed of local search heuristics. Our proposed method feeds the solution of one algorithm phase to the other one, until no further improvements can be made. The applicability of this method is demonstrated experimentally on the galaxy instance set, resulting in currently best known solutions for most of the considered instances.

We consider the traveling tournament problem (TTP) and the traveling umpire problem (TUP). In TTP, the task is to design a double round-robin schedule, where no two teams play against each other in two consecutive rounds, and the total travel distance is minimized. In TUP, the task is to find an assignment of umpires for a given tournament such that every umpire handles at least one game at every team's home venue and an umpire neither visits a venue nor sees a team (home or away) too often. The task is to minimize the total distance traveled by the umpires. We present a combined approximation for this problem, when the number of umpires is odd. We therefore first design an approximation algorithm for TTP and then show how to define an umpire assignment for this tournament such that a constant-factor approximation for TUP is guaranteed.

In the veneer cutting industry tree trunks are peeled into thin veneer strips which are cut, glued together, and pressed into bentwood pieces for seats, backrests, etc. In this work, a model for optimizing the inherent cutting problem with respect to resource efficiency is presented. Especially the heterogeneous quality of the wood renders existing models for classic cutting stock problems useless and calls for a new modeling approach. By means of the model presented in this paper, the problem is solved to optimality for real-world instances in reasonable time and applicable solutions are generated. Furthermore, in order to deal with uncertainties in the wood quality, the approach of robust optimization is applied to the problem. Robust optimization is an important tool to deal with uncertainties in the formulation of mathematical optimization models. Different concepts of robustness have been provided in the literature, one of which is the concept of minmax robust efficiency for uncertain multi-objective optimization problems. The concept of minmax robust efficiency is applied to a simplified version of the problem, robust efficient solutions are calculated, and the paper concludes with the discussion of the benefit of these solutions.

( ( 1 - 1 e k ) β ) -approximation for k-SAP if the single bin problem admits a β-approximation. ( 1 - 1 e k ) -approximation for k-SAP if the single bin problem admits an FPTAS.For k 2 , our algorithm beats the upper bound of ( 1 - 1 e ) known for other special cases of SAP. We consider a variant of the separable assignment problem (SAP). In the classic version of SAP, we are given a set of bins and a set of items to pack into the bins together with a profit p i , j for assigning item i to bin j. Each bin j has a separate packing constraint, i.e., only certain subsets of the items fit into bin j. The objective is to find an assignment of a subset of the items to the bins such that the packing constraints of all bins are satisfied, no item is assigned to more than one bin, and the total profit is maximized. As an important special case, this problem contains the maximum generalized assignment problem (GAP).It is known that, given a β-approximation algorithm for the single bin subproblem (i.e., the problem of finding the most profitable packing for a single bin), it is possible to obtain a ( ( 1 - 1 e ) β ) -approximation for SAP using randomized rounding. If the single bin subproblem admits an FPTAS, one can obtain a ( 1 - 1 e ) -approximation. This is best possible in the sense that there exist special cases of SAP which do not admit polynomial-time approximation algorithms with an approximation factor better than ( 1 - 1 e ) unless NP DTIME ( n O ( log log n ) ) .In this paper, we consider the case of SAP where each item may be assigned at most k 1 times (but at most once to each bin) and present a ( ( 1 - 1 e k ) β ) -approximation algorithm for this case under the assumption that the single bin subproblem admits a β-approximation algorithm. If the single bin subproblem admits an FPTAS, we obtain a ( 1 - 1 e k ) -approximation, which shows that, for k 2 , the problem admits approximation algorithms that beat the upper bound of ( 1 - 1 e ) known for other special cases of SAP.

In light of various environmental problems and challenges concerning resource allocation, the utilisation of renewable resources is increasingly important for the efficient use of raw materials. Therefore, cascading utilisation (i.e., the multiple material utilisations of renewable resources prior to their conversion into energy) and approaches that aim to further increase resource efficiency (e.g., the utilisation of by-products) can be considered guiding principles. This paper therefore introduces the Special Volume “Improved Resource Efficiency and Cascading Utilisation of Renewable Materials”. Because both research aspects, resource efficiency and cascading utilisation, belong to several disciplines, the Special Volume adopts an interdisciplinary perspective and presents 16 articles, which can be divided into four subjects: Innovative Materials based on Renewable Resources and their Impact on Sustainability and Resource Efficiency, Quantitative Models for the Integrated Optimisation of Production and Distribution in Networks for Renewable Resources, Information Technology-based Collaboration in Value Generating Networks for Renewable Resources, and Consumer Behaviour towards Eco-friendly Products. The interdisciplinary perspective allows a comprehensive overview of current research on resource efficiency, which is supplemented with 15 book reviews showing the extent to which textbooks of selected disciplines already refer to resource efficiency. This introductory article highlights the relevance of the four subjects, presents summaries of all papers, and discusses future research directions. The overall contribution of the Special Volume is that it bridges the resource efficiency research of selected disciplines and that it presents several approaches for more environmentally sound production and consumption.

In this paper, we discuss the problem of finding an optimal schedule for the German Basketball League BBL for the 2011-2012 season. A major issue that we address is that most of the games take place in multipurpose arenas that are also used for other events and are thus not always available. In addition, we must minimize the number of successive home or away games, assign the most interesting games to television broadcasting slots, minimize the distance teams must drive on a newly established derby day, and meet team requests for home or away games and to play specific teams.We present several algorithmic approaches and show how these models fit the BBL's requirements. In this process, we prove that the classic models, which the BBL had applied previously and many other leagues still apply, are too limited to meet these requirements. We show that canonical schedules do not have the desired properties. We proved to the BBL that mirrored schedules cannot meet its needs, and thus convinced it to use nonmirrored schedules for the first time in its history. The BBL's requirements are typical of sports leagues; therefore, the approaches presented are also applicable to many other sports leagues. We implemented our approach in scheduling software that we developed for the BBL and applied to its 2011-2012 season scheduling.

We consider a generalization of the maximum generalized assignment problem. We relax the hard constraints for the bin capacities, and introduce for every bin a cost function that is convex in the total load on this bin. These costs are subtracted from the profits of assigned items, and the task is to find an assignment maximizing the resulting net profit. We show that even restricted cases of this problem remain strongly NP-complete, and identify two cases that can be solved in strongly polynomial time. Furthermore, we present a $$(1-1/e)$$ -approximation algorithm for the general case. This algorithm uses a configuration based integer programming formulation for a randomized rounding procedure. In order to turn the rounded solution into a feasible solution, we define appropriate estimators that linearize the convex costs.

The Traveling Tournament Problem (TTP) is a complex problem in sports scheduling whose solution is a schedule of home and away games meeting specific feasibility requirements, while minimizing the total distance traveled by all the teams. A recently-developed "hybrid" algorithm, combining local search and integer programming, has resulted in best-known solutions for many TTP instances. In this paper, we tackle the TTP from a graph-theoretic perspective, by generating a new "canonical" schedule in which each team's three game road trips match up with the underlying graph's minimum-weight P-3-packing. By using this new schedule as the initial input for the hybrid algorithm, we develop tournament schedules for five benchmark TTP instances that beat all previously-known solutions.